Lévy decoupled random walks

✍ Scribed by Miguel A. Ré; Carlos E. Budde; Domingo P. Prato

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 185 KB
• Volume: 323
• Category: Article
• ISSN: 0378-4371
• DOI: 10.1016/s0378-4371(03)00016-5

✦ Synopsis

A di usion model based on a continuous time random walk scheme with a separable transition probability density is introduced. The probability density for long jumps is proportional to x -1- (a LÃ evy-like probability density). Even when the probability density for the walker position at time t; P(x; t), has not a ÿnite second moment when 0 ¡ ¡ 2, it is possible to consider alternative estimators for the width of the distribution. It is then found that any reasonable width estimator will exhibit the same long-time behaviour, since in this limit P(x; t) goes to the distribution L (x=t ), a LÃ evy distribution. The scaling property is veriÿed numerically by means of Monte Carlo simulations. We ÿnd that if the waiting time density has a ÿnite ÿrst moment then = 1= , while for densities with asymptotic behaviour t -1-ÿ with 0 ¡ ÿ ¡ 1 ("long tail" densities) it is veriÿed that =ÿ= . This scaling property ensures that any reasonable estimator of the distribution width will grow as t in the long-time limit. Based on this long-time behaviour we propose a generalized criterion for the classiÿcation in superdi usive and subdi usive processes, according to the value of .